On the Dependence between Quantiles and Dispersion Estimators

ESSEC WORKING PAPER 1807

92 Pages Posted: 6 Oct 2019

See all articles by Marcel Bräutigam

Marcel Bräutigam

ESSEC Business School; Sorbonne University

Marie Kratz

ESSEC Business School - Information & Decision Sciences Department

Date Written: December 2018

Abstract

In this study, we derive the joint asymptotic distributions of functionals of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile) and functionals of measure of dispersion estimators (the sample standard deviation, sample mean absolute deviation, sample median absolute deviation) - assuming an underlying identically and independently distributed sample. Additionally, for location-scale distributions, we show that asymptotic correlations of such functionals do not depend on the mean and variance parameter of the distribution. Further, we compare the impact of the choice of the quantile estimator (sample quantile vs. parametric location-scale quantile) in terms of speed of convergence of the asymptotic covariance and correlations respectively. As application, we show in simulations a good finite sample performance of the asymptotics. Further, we show how the theoretical dependence results can be applied to the most well-known risk measures (Value-at-Risk, Expected Shortfall, expectile). Finally, we relate the theoretical results to empirical findings in the literature of the dependence between risk measure prediction (on historical samples) and the estimated volatility.

Keywords: Asymptotic Distribution, Sample Quantile, Measure of Dispersion, Non-linear Dependence, VaR, ES, Correlation

JEL Classification: C13, C14, C30, C58, C69, G32

Suggested Citation

Bräutigam, Marcel and Kratz, Marie, On the Dependence between Quantiles and Dispersion Estimators (December 2018). ESSEC WORKING PAPER 1807 , Available at SSRN: https://ssrn.com/abstract=3459495 or http://dx.doi.org/10.2139/ssrn.3459495

Marcel Bräutigam (Contact Author)

ESSEC Business School ( email )

3 Avenue Bernard Hirsch
CS 50105 CERGY
CERGY, CERGY PONTOISE CEDEX 95021
France

Sorbonne University ( email )

UFR 927, 4 Place Jussieu
Paris, F-75252
France

Marie Kratz

ESSEC Business School - Information & Decision Sciences Department ( email )

Avenue Bernard Hirsch B.P. 50105
Cergy-Pontoise (Paris), 95021
France

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